GENERALIZED QUATERNIONS SERRET-FRENET AND BISHOP FRAMES SERRET-FRENET VE BISHOP ÇATILARI

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1 Sy 9, Arlk 0 GENERALIZED QUATERNIONS SERRET-FRENET AND BISHOP FRAMES Erhn ATA*, Ysemin KEMER, Ali ATASOY Dumlupnr Uniersity, Fculty of Science nd Arts, Deprtment of Mthemtics, KÜTAHYA, et@dpu.edu.tr ABSTRACT Serret-Frenet nd Prllel-Trnsport frme re produced with the help of the generlized quternions gin by the method in [4]. Keywords: Generlized quternion, Serret-Frenet frme, Bishop frme. GENELLETRLM KUATERNYONLARIN SERRET-FRENET VE BISHOP ÇATILARI ÖZET Serret-Frenet e Prlel tm çtlr, genelletirilmi kuterniyonlr yrdmyl yine [4]te erilen metot ile oluturulmutur.. INTRODUCTION The Frenet-Serret formuls describe the kinemtic properties of prticle which moes long continuous, differentible cure in Eucliden spce ³ or Minkowski spce ³. These formuls he common re of usge in mthemtics, physics (especilly in reltie theory), medicine, computer grphics nd such fields. It is known by especilly mthemticins nd physicists tht ny unit (split) quternion corresponds to rottion in Eucliden nd Minkowski spces. For detiled informtion it is referred to [], [] nd []. The rottions re expressed by quternions tht is becuse the geodesic cures in unit (split) quternion spce S³ cn not be expressed by using Euler ngles [4].. PRELIMINARIES Our first gol is to define moing coordinte frmes tht re ttched to cure in D spce... Frenet-Serret frmes: The Frenet-Serret frme (see, [5], [6] nd [7]) is defined s follows: Let (t) be ny thrice-differentible spce cure with non-nishing second deritie. We cn choose this locl coordinte system to be the Frenet-Serret frme consisting of the tngent T (t), the binorml B(t), nd the principl norml N(t) ectors t point on the cure gien by 9

2 Sy 9, Arlk 0 '(t) '(t) ''(t) T(t)=, B(t)=, N(t)= B(t) T(t). '(t) '(t) ''(t) (.) The Frenet-Serret frme (lso known s the Frenet frme) obeys the following differentil eqution in the prmeter t: T'(t) 0 ( t) 0 T(t) N'(t) (t) ( t) 0 ( t) N(t) B'(t) 0 ( t) 0 B(t) (.) where t ' t is sclr mgnitude of the cure deritie (often reprmetrized to be unity, so tht t becomes the rc-length s), the intrinsic geometry of the cure is embodied in the sclr curture () t nd the torsion () t. In principle these quntities cn be clculted in terms of the prmetrized or numericl locl lues of () t nd its first three derities s follows: '( t) ''( t) () t '( t) '( t) ''( t) '''( t) () t '( t) ''( t) (.) If non-nishing curture nd torsion re gien s smooth function of t, the system of equtions cn be integrted theoreticlly to find the unique numericl lues of the corresponding spce cure (). t.. Prllel Trnsport Frmes: Intuitiely, the Frenet frme's norml ector N lwys points towrd the center of the osculting circle [8]. Thus, when the orienttion of the osculting circle chnges drsticlly or the second deritie of the cure becomes ery smll, The Frenet frme behes errticlly or my become undefined. Prllel Trnsport Frmes: The Prllel Trnsport frme or Bishop frme is n lterntie pproch to defining moing frme tht is well defined een when the cure hs nishing second deritie. We cn prllel trnsport n orthonorml frme long cure simply by prllel trnsporting ech component to the frme. The prllel trnsport frme is bsed on the obsertion tht, while Tt () for gien cure model is unique, we my choose ny conentient rbitrry bsis N(), t N() t for the reminder of the frme, s long s it is in the norml plne perpendiculr to Tt () t ech point. If the derities of N(), t N() t depend only on Tt () nd not on ech other, we cn mke N () t nd N () t 0

3 Sy 9, Arlk 0 ry smoothly throughout the pth regrdless of the curture. We therefore he the lterntie frme equtions T'(t) 0 k (t) ( t) k( t) T N'(t) (t) k( t) N(t) N'(t) k () t B(t) (.4) One cn show (see, [0]) tht k k, (.5) k ( t) rctn, k (.6) d t () t dt (.7) so tht k nd k effectiely correspond to Crtesin coordinte system for the polr coordintes, with dt. The orienttion of the prllel trnsport frme includes the rbitrry choice of integrtion constnt 0, which disppers from (nd hence the Frenet frme) due to the differentition.. GENERALIZED QUATERNION FRAMES Definition.. The set H, q 0 i j k : 0,,,,, hing bsis, i, j, k with the following properties: is ector spce oer i, j, k ij ji k jk kj i ki ik j Eery element of the set H is clled generlized quternion [9]. Definition.. A generlized quternion frme is defined s unit-lenght generlized quternion q 0i j k nd is chrcterized by the following properties: Two generlized quternions q nd p obey the following multipliction rule, qp b b b b b b b b i The conjugte of q is defined s 0 0 b b b b j b b b bk q i j k. 0 (.) A unit-length generlized quternion's norm is defined s: N qq qq q 0.

4 Sy 9, Arlk 0 Eery possible rottion R ( specil generlized orthogonl mtrix) cn be constructed from either of two relted generlized quternions, q 0i j k or q 0i j k, using the trnsformtion lw: qwq Rw q w q R i ij w j j where wi jk is generlized pure quternion. We compute Rij directly from (.) R All rows of this mtrix expressed in this form re orthogonl but not orthonorml. Diiding first, second nd lst column by, nd, respectiely we get R (.) All rows of this rottion mtrix expressed in this form re orthonorml nd crete roof. The qudrtic form (.) for generl orthonorml frme coincides with Frenet nd prllel trnsport frmes. Specil cses: (i) For, the generlized quternion lgebr H coincides with the rel quternion lgebr H. In this cse the rottion mtrix R becomes R These mtrices form the three-dimensionl specil orthogonl group SO(). Since the mtrix R cn be obtined by the unit quternions q nd q, there re two unit quternions for eery rottion in Eucliden spce. (ii) For,, the generlized quternion lgebr H coincides with the split quternion lgebr H. In this cse the rottion mtrix R becomes

5 Sy 9, Arlk 0 R These mtrices form the three-dimensionl specil orthogonl group SO(,). Similrly the mtrix R cn be obtined by the unit split quternions q nd -q, there re two unit timelike quternions for eery rottion in Minkowski -spce. The equtions obtined s result of this coincidence re quternion lued liner equtions. If we derie the rows eqution of (.) respectiely, then we obtin following results; 0 0 dt 0 Aq 0 dn B q 0 0 (.) 0 0 db 0 Cq. 0.. Generlized Quternion Frenet Frme Eqution: The Frenet equtions themseles must tke the form Aq T N Bq N T B C q B N where 0 b0 b b b0 c0 c c c q. d0 d d d e e e e 0 (.4) (.5) (.6)

6 Sy 9, Arlk 0 Therefore; with the help of (.4),(.5) nd (.6) we obtin the following equtions: b b b b c c c c d d d e e 0 e 0 e 0 0 (.7) b b b b c c c c e ee e (.8) b b b b c c 0 c 0 c d d d d e e e e 0 0 (.9) b b b b c c 0 c 0 c d d d d e e e e 0 (.0) Finlly, we get b, b, b 0, b, c0, c 0, c, c 0, d, d, d 0, d, e0, e 0, e, e 0. Therefore, the generlized quternion Frenet frme eqution: q. 0 0 Specil cse: (i) For we get the rel quternion Frenet frme eqution 4

7 Sy 9, Arlk 0 for quternion lgebr H. q (ii) For, we get the split quternion Frenet frme eqution q with the split quternion lgebr H... Prllel-Trnsport Generlized Quternion Frme Eqution Similrly, it cn be esily shown tht prllel trnsport frme system with N (), t T (), t N () t (in tht order) corresponded to columns of eqution (.) is completely equilent to the following prlleltrnsport generlized quternion frme eqution: Bq T' knkn (.) Aq N' kt (.) Cq N' kt (.) where 0 b0 b b b0 c0 c c c q. d0 d d d e0 e e e Therefore; with the help of (.),(.) nd (.) we obtin the following equtions: b00 b b b c00 c c c e e 0 e 0 e 0 k 0 k 0 (.4) b b b b c c 0 c 0 c d00 d e00 e e e k 0 k 0 (.5) 5

8 Sy 9, Arlk 0 b b b b c c c c d d d e e 0 e 0 e 0 k 0 (.6) b b b b c c 0 c 0 c d d d d e e e e k 0 (.7) Finlly, we get b, b k, b 0, b k, k k c0, c 0, c, c 0, d, d k, d 0, d k, k k e0, e 0, e, e 0. Therefore, the generlized quternion prllel-trnsport frme eqution: 0 k 0 k 0 k k 0 q. 0 k 0 k k k Specil cse: (i) For we get the rel quternion prllel-trnsport frme eqution 0 0 k 0 k0 k 0 k 0 q. 0 k 0 k k 0 k 0 for quternion lgebr H. (ii) For, we get the split quternion Frenet frme eqution with the split quternion lgebr H. 0 0 k 0 k0 k 0 k 0 q. 0 k 0 k k 0 k 0 6

9 Sy 9, Arlk 0.. Conclusion While the rottions cn be expressed by using the Euler ngles, the rottions between the geodesic cures in the unit (split) quternion spce cn not be obtined by the Euler ngles. In ddition, it is necessry to sole nine-component eqution for rottion or trnsltion mde by using the Euler ngles. Wheres, insted of this, it cn be mde by unit (split) quternion. REFERENCES [] Inoguchi, J., Timelike surfces of constnt men curture in Minkowski - spce, Tokyo J. Mth. () 4-5, 998. [] Nien, I., The roots of quternion, Amer. Mth. Monthly 449(6) 86-88, 94. [] Özdemir, M., Ergin A. A., Rottions with timelike quternions in Minkowski -spce, J. Geom. Phys. 56-6, 006 [4] Hnson, A. J., Quternion Frenet Frmes: Mking Optiml Tubes nd Ribbons from Cures, Tech. Rep. 407, Indin Un. Computer Science Dep., 994. [5] Eisenhrt, L. P., A Tretise on the Differentil Geometry of Cures nd Surfces, Doer, New York, 960, Originlly published in 909. [6] Flnders, H., Differentil Forms with Applictions to Physicl Sciences, Acdemic Press, New York, 96. [7] Gry, A., Modern Differentil Geometry of Cures nd Surfces, CRC Press, Inc., Boc Rton, FL, 99. [8] Struik, D. J., Lectures on Clssicl Differentil Geometry, Addison-Wesley, 96 [9] Öztürk, U., Hcsliholu, H. H., Yyl, Y., Koç Öztürk, E. B., Dul Quternion Frmes, Commun. Fc. Sci. Uni. Ank. Series A 59() 4 50, 00 [0] Bishop, R. L., There is more thn one wy to frme cure, Amer. Mth. Monthly 8(), 46-5, Mrch

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